Could you please tell me what is the difference between confidence interval and confidence region in the following sense?

For example, we have s multiple linear regression model. For individual confidence intervals, we use $t$-statistics to find individual confidence intervals for regression parameters but I found in many books that when authors write confidence intervals based on $F$ distribution,they call it confidence region not the intervals.

What does it mean by region and how does it relate to confidence intervals, specifically in view of multiple regression?


1 Answer 1


As @NBrouwer says, a confidence interval is for an individual variable, so it is a (one-dimensional) interval. This is the case for e.g. the confidence interval for an individual regression coefficent.

However, if you build 'confidence intervals' for more than one variable at a time, i.e. for a multivariate parameter $\beta=(\beta_0, \beta_1, \dots \beta_n)$ then you get a region in an (n+1)-dimensional space. In two dimensions this could be a rectangular region, an ellips, or another shape.

The Bonferroni correction e.g. requires you (for the two variable case) for a 0.95 confidence level to construct two intervals - one for each dimension - using a 0.975 confidence level. The confidence region then looks like $\{(x,y) | \bar{x}_L \le x \le \bar{x}_H \& \bar{y}_L \le y \le \bar{y}_H \}$ which defines a rectangular region (the 'bar' means a fixed value and the subscripts L and H mean Low and High). Such a rectangular region could be seen as a (two-dimensional) interval.

For confidence 'intervals' based on statistics like a $\chi^2$-statistic (in 2 dimensions) your region will be an ellips (see also How to find the maximum axis of ellipsoid given the covariance matrix? and Chi-Square-Test: Why is the chi-squared test a one-tailed test? - where the equation for an ellips can be seen in the definition of the $X^2$).

Similar for the F-statistic; you build an interval for all the coefficients jointly, so it is a multidimensional region.

So a one-dimensional confidence region is a confidence interval. Rectangular confidence regions might be called confidence intervals, depending on your definition of an (multi-dimensional) interval. Or ''confidence region'' is the more general name, ''confidence inteval'' is a special case of a ''confidence region''.


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